New Sextic Fields Unramified Outside of {∞, 2, 5}

Discriminant class: −10

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
1. 3_{i} 9_{k} 7_{+} -2 -5 10 2^{2} 6_{i} −1 ∗ 1^{5} 7_{+} 5∗  32 42 5 5 6 5 3 32
S_{6} 25 -64 38 1^{2} 3_{−} 2∗ 2∗ 32 5 42 6 33 5 42 32
2. 1_{k} 9_{i} 7_{+} -2 0 10 1^{4} 6_{+} ∗ −1 1^{5} 7_{+} 5∗  6 42 5 5 32 5 33 6
S_{6}  -15 -8 16 1^{2} 3_{i} −2 −2  6 5 42 32 3 5 42 6
3. 1_{k} 11_{i} 7_{+} 0 0 0 1^{4} 8_{+} ∗  1^{5} 7_{+} 5∗  32 33 42 33 32 42 5 4
S_{6}  -15 -8 10 1^{2} 3_{i} −2 −2  6 42 5 6 33 5 42 32
4. 3_{i} 11_{k} 7_{+} 0 0 0 1^{6} 11_{k} −2∗  1^{5} 7_{+} 5∗  6 3 42 3 6 42 5 4
S_{6}  10 -8 10  32 42 5 32 3 5 42 6

Discriminant class: −5

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
5. 3_{i} 6_{i} 9_{−}  2 10 10 1^{4} 4_{+} ∗  1^{5} 9_{−} 5  5 3 6 32 32 32 33 22
S_{6}  15  2 14 1^{2} 2_{i} −1 −1  32 32 42 3 42 4 6 33
6. 1_{k} 8_{k} 9_{−}  0 -15 10 1^{6} 8_{k} −∗  1^{5} 9_{−} 5  5 33 32 6 6 6 3 22
S_{6}  5 -14 -5  6 6 42 33 42 4 32 3
7. 3_{i} 10_{i} 9_{−} -3 25 -40 1^{4} 10_{i} −1  2 1^{5} 9_{−} 5  42 5 32 32 4 4 5 5
S_{6}  160 -96 288 2 0_{+} ∗  ∗  6 32 5 3 33 32 6 22
8. 3_{i} 10_{i} 9_{−}  1  5 10 1^{4} 10_{i} −1  2 1^{5} 9_{−} 5  5 5 6 4 4 6 42 5
S_{6}  75 113 63 2 0_{+} ∗  ∗  2 6 5 5 33 6 6 42
9. 1_{k} 12_{k} 9_{−} -2 -5 20 1^{4} 9_{+} 2 −1 1^{5} 9_{−} 5  42 5 6 6 4 4 5 5
S_{6}  -100 -356 -238 1^{2} 3_{k} −2∗ −2∗  32 6 5 33 3 6 32 22
10. 1_{k} 12_{k} 9_{−} -2 -5  0 1^{4} 9_{+} 2 −1 1^{5} 9_{−} 5  5 5 32 4 4 32 42 5
S_{6}  20 12 -14 1^{2} 3_{k} −2∗ −2∗  222 32 5 5 3 32 32 42
11. 3_{i} 14_{i} 9_{−} -2 25 -60 1^{4} 11_{+} 2∗  2∗ 1^{5} 9_{−} 5  5 5 4 32 32 4 3 42
S_{6}  185 -54 133 1^{2} 3_{i} −2 −2  32 4 3 33 5 222 222 42
12. 1_{k} 14_{k} 9_{−} -2 -15 40 1^{4} 11_{−} 2∗  2∗ 1^{5} 9_{−} 5  5 5 4 6 6 4 33 42
S_{6}  -10 -372 -346 1^{2} 3_{i} −2 −2  6 4 33 3 5 2 2 42

Discriminant class: −2

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
13. 3_{i} 11_{k} 8_{+} 0  0 0 1^{4} 8_{+} ∗  1^{5} 8_{+} ∗  5 32 5 4 3 33 4 32
S_{6}  2  25 40 50 1^{2} 3_{k} −2∗ −2∗  32 6 5 5 32 32 33 6
14. 1_{k} 11_{i} 8_{+} -2 -10 0 1^{4} 11_{i} −2∗ 2∗ 1^{5} 8_{+} ∗  5 32 42 32 33 5 32 4
S_{6}  25 10 -20 2 0_{+} ∗  ∗  32 6 42 5 32 6 42 32
15. 1_{k} 11_{i} 8_{+} 2 -5 0 1^{6} 11_{i} −2  1^{5} 8_{+} ∗  5 6 5 4 33 3 4 6
S_{6}  -5 -22 -9  6 32 5 5 6 6 3 32
16. 3_{i} 13_{k} 8_{+} 2 -10 0 1^{4} 11_{−} 2∗ −2∗ 1^{5} 8_{+} ∗  5 6 42 6 3 5 6 4
S_{6}  0 -160 480 1^{2} 2_{i} −∗ −∗  6 32 42 5 6 32 42 6

Discriminant class: 1

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
17. 2_{−} 14_{−} 8_{+} 0 0 -20 1^{4} 11_{i} −2 2 1^{5} 8_{+} 1  42 5 5 5 42 5 22 42
A_{6} 3 -5 4 10 1^{2} 3_{i} −2 −2 33 5 33 42 5 5 33 5
18. 2_{−} 14_{−} 8_{+} -2 15 0 1^{4} 11_{−} 2 −2 1^{5} 8_{+} 1  42 5 5 5 42 5 22 42
A_{6}  3  50 -4 -82 1^{2} 3_{+} 2  2  3 5 3 42 5 5 3 5

Discriminant class: 2

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
19. 2_{−} 5_{−} 8_{+} 3  5  5 1^{3} 2_{+} ∗  1^{5} 8_{+} ∗  6 42 6 6 5 6 42 6
S_{6}  10 12 -4 1^{2} 3_{−} 2∗  2∗  3 6 5 6 22 32 6 6
20. 2_{−} 11_{−} 8_{+} 0  0  0 1^{6} 11_{−} 2  1^{5} 8_{+} ∗  32 42 32 32 5 32 42 32
S_{6}  2  0 -40 -50  33 32 5 32 22 6 32 32
21. 2_{−} 13_{−} 8_{+} -2 10 -40 1^{4} 11_{i} −2 −2∗ 1^{5} 8_{+} ∗  6 33 6 32 33 6 42 32
S_{6} 90 -116 52 1^{2} 2_{i} −1 −1 42 6 5 32 5 32 32 32
22. 2_{−} 13_{−} 8_{+} -2 -15 40 1^{4} 11_{i} −2∗ −2 1^{5} 8_{+} ∗  32 3 32 6 3 32 42 6
S_{6}  40 -72 -196 1^{2} 2_{i} −∗ −∗  42 32 5 6 5 6 6 6

Discriminant class: 5

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
23. 2_{−} 12_{+} 9_{−}  0 -25  0 1^{4} 10_{k} −1 2∗ 1^{5} 9_{−} 5  32 32 5 6 6 3 6 5
S_{6}  175 -80 -375 1^{2} 2_{i} −∗ −∗  42 6 33 32 6 4 33 42
24. 2_{−} 12_{+} 9_{−} -2 -5  0 1^{4} 9_{+} 2∗ −1 1^{5} 9_{−} 5  6 6 5 32 32 33 32 5
S_{6}  20 32 -104 1^{2} 3_{+} 2  2  42 32 3 6 32 4 3 42
25. 2_{−} 14_{+} 9_{−} -2 -25  0 1^{4} 11_{i} −2  2 1^{5} 9_{−} 5  32 32 42 4 32 42 32 42
S_{6}  200 360 180 1^{2} 3_{k} −2∗ −2∗  3 6 42 4 6 6 5 5
26. 2_{−} 14_{+} 9_{−}  0 -10 -20 1^{4} 11_{−} 2 −2 1^{5} 9_{−} 5  6 6 42 4 6 42 6 42
S_{6}  -45 -28 -20 1^{2} 3_{−} 2∗  2∗  33 32 42 4 32 32 5 5
27. 2_{−} 14_{+} 9_{−}  0 -10 -80 1^{4} 11_{i} −2 −2∗ 1^{5} 9_{−} 5  6 6 3 32 6 5 6 42
S_{6}  -170 -352 -20 1^{2} 3_{k} −2∗ −2∗  42 4 5 222 32 32 42 42
28. 2_{−} 14_{+} 9_{−} -2 -5  0 1^{4} 11_{k} −2∗ −2 1^{5} 9_{−} 5  32 32 33 6 32 5 32 42
S_{6}  -30 -28 166 1^{2} 3_{i} −2 −2  42 4 5 2 6 6 42 42

Discriminant class: 10

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
29. 2_{−} 11_{−} 5_{+} -1 5 0 1^{4} 11_{−} 2 −2 1^{5} 5_{+} 5∗  5 6 6 5 32 32 32 6
S_{6}  0 8 -8 2 0_{+} ∗  ∗  42 42 42 42 6 5 6 6
30. 2_{−} 11_{−} 5_{+} 2 5 0 1^{4} 8_{−} ∗  1^{5} 5_{+} 5∗  5 6 6 42 6 32 6 32
S_{6}  -10 -8 -6 1^{2} 3_{+} 2  2  5 5 42 5 6 42 6 4

Discriminant class: 10 (continued)

∞  2 5 a_{1} a_{2} a_{3}  Over p = 2  Over p = 5 3 7 11 13 17 19 23 29
G  h  a_{4} a_{5} a_{6} f^{e} c_{w} d  s f^{e} c_{w} d s 31 37 41 43 47 53 59 61
31. 2_{−} 11_{−} 5_{+} -2 5 0 1^{6} 11_{−} 2∗  1^{5} 5_{+} 5∗  5 32 32 42 32 6 32 6
S_{6}  -5 10 -5  5 5 42 5 32 42 32 4
32. 2_{−} 13_{−} 5_{+} -2 0 0 1^{4} 11_{i} −2 2 1^{5} 5_{+} 5∗  5 32 32 5 6 6 6 32
S_{6}  5 10 -10 1^{2} 2_{i} −∗ −∗  42 42 42 42 32 5 32 32